42 2. FOUR IMPORTANT LINEAR PDE

THEOREM 16 (Uniqueness). There exists at most one solution u ∈

C2(

¯

U ) of (46).

Proof. Assume ˜ u is another solution and set w := u − ˜. u Then Δw = 0 in

U, and so an integration by parts shows

0 = −

U

wΔw dx =

U

|Dw|2

dx.

Thus Dw ≡ 0 in U, and, since w = 0 on ∂U, we deduce w = u − ˜ u ≡ 0 in

U.

b. Dirichlet’s principle. Next let us demonstrate that a solution of the

boundary-value problem (46) for Poisson’s equation can be characterized as

the minimizer of an appropriate functional. For this, we deﬁne the energy

functional

I[w] :=

U

1

2

|Dw|2

− wf dx,

w belonging to the admissible set

A := {w ∈

C2(

¯

U ) | w = g on ∂U}.

THEOREM 17 (Dirichlet’s principle). Assume u ∈

C2(

¯

U ) solves (46).

Then

(47) I[u] = min

w∈A

I[w].

Conversely, if u ∈ A satisﬁes (47), then u solves the boundary-value problem

(46).

In other words if u ∈ A, the PDE −Δu = f is equivalent to the statement

that u minimizes the energy I[ · ].

Proof. 1. Choose w ∈ A. Then (46) implies

0 =

U

(−Δu − f)(u − w) dx.

An integration by parts yields

0 =

U

Du · D(u − w) − f(u − w) dx,